Question: Factor the following expression: $3$ $x^2$ $-1$ $x$ $-24$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(3)}{(-24)} &=& -72 \\ {a} + {b} &=& & & {-1} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-72$ and add them together. Remember, since $-72$ is negative, one of the factors must be negative. The factors that add up to ${-1}$ will be your ${a}$ and ${b}$ When ${a}$ is ${8}$ and ${b}$ is ${-9}$ $ \begin{eqnarray} {ab} &=& ({8})({-9}) &=& -72 \\ {a} + {b} &=& {8} + {-9} &=& -1 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {3}x^2 +{8}x {-9}x {-24} $ Group the terms so that there is a common factor in each group: $ ({3}x^2 +{8}x) + ({-9}x {-24}) $ Factor out the common factors: $ x(3x + 8) - 3(3x + 8) $ Notice how $(3x + 8)$ has become a common factor. Factor this out to find the answer. $(3x + 8)(x - 3)$